# Sign Patterns of Inverse Doubly-Nonnegative Matrices

## TL;DR

This paper characterizes the sign patterns of inverse doubly-nonnegative matrices, providing necessary and sufficient conditions and revealing unique sign patterns for matrices with tree-structured graphs.

## Contribution

It introduces a complete criterion for sign patterns of inverse doubly-nonnegative matrices and describes their structure for tree graphs, advancing understanding of their inverse properties.

## Key findings

- A necessary and sufficient condition for sign patterns of inverse doubly-nonnegative matrices.
- Unique sign pattern characterization for matrices with tree graphs.
- Sign pattern expressible via two-coloring of the graph.

## Abstract

The sign patterns of inverse doubly-nonnegative matrices are examined. A necessary and sufficient condition is developed for a sign matrix to correspond to an inverse doubly-nonnegative matrix. In addition, for a doubly-nonnegative matrix whose graph is a tree, the inverse is shown to have a unique sign pattern, which can be expressed in terms of a two-coloring of the graph.